Sliding mode control method

ABSTRACT

According to a sliding mode control method, a hyperplane for a sliding mode control process is established with a linear function having as variables a plurality of state quantities of an object to be controlled. The state quantities are converged onto the hyperplane, and also converged toward a balanced point on the hyperplane while the state quantities are being converged onto the hyperplane, thereby to control the state quantities at target state quantities represented by the balanced point. The hyperplane are variably established depending on the manner in which the state quantities are converged onto the hyperplane.

BACKGROUND OF THE INVENTION

1. Field of the Invention:

The present invention relates to a sliding mode control method.

2. Description of the Prior Art:

Heretofore, a PID (proportional plus integral plus derivative) controlprocess or a feedback control process employing an optimum regulator hasgenerally been used to control a plurality of state quantities of anobject to be controlled (e.g., the displacement of an object to becontrolled and a rate of change of the displacement).

However, it is difficult for the conventional control process such asthe PID control process or the like to converge the state quantitieswith sufficient quick response or stability against disturbances,characteristic changes of the object to be controlled, etc. For theoptimum regulator, it is necessary to construct a model of the object tobe controlled. As an error (a model error) between the model and theactual object to be controlled increases owing to dynamic characteristicchanges of the object to be controlled, it is difficult to maintainsufficient quick response or stability for the convergence of the statequantities.

In view of the above conventional drawbacks, it has recently beenpracticed to control state quantities of an object to be controlled in asliding mode control process according to the modern control technology.

The sliding mode control process is a feedback control process ofvariable structure. According to the sliding mode control process, ahyperplane (see FIG. 7 of the accompanying drawings) expressed by alinear function which has as its variables a plurality of statequantities of an object to be controlled is defined in advance, and thestate quantities are converged onto the hyperplane under high-gaincontrol. Furthermore, while the state quantities are being convergedonto the hyperplane, the state quantities are converged toward a givenbalanced point (a point corresponding to target state quantities) by anequivalent control input.

The sliding mode control process has excellent characteristics in thatonce the state quantities are converged onto the hyperplane, the statequantities can stably be converged toward the balanced point withoutbeing substantially subject to effects of disturbances, etc. Therefore,it is possible to increase the quick response and stability ofconvergence of the state quantities against disturbances, etc.

For controlling the state quantities of the object to be controlledaccording to the sliding mode control process, it is desirable toincrease the quick response and stability of convergence of the statequantities onto the hyperplane and also quick response and stability ofconvergence of the state quantities toward the balanced point on thehyperplane for the purpose of sufficiently maintaining the quickresponse and stability of convergence of the state quantities.

According to the inventor's understanding, if the state quantities inthe sliding mode control process represent two values, then when thestate quantities are already substantially converged onto thehyperplane, the stability of convergence of the state quantities towardthe balanced point on the hyperplane is greater and the time requiredfor the state quantities to converge toward the balanced point isshorter (the quick response is greater) as the gradient of thehyperplane is greater. When the state quantities are not still convergedonto the hyperplane, the state quantities can be converged relativelyquickly onto the hyperplane while being prevented from sufferingoscillatory responses with respect to the hyperplane if the gradient ofthe hyperplane is smaller (the quick response can be increased).

According to the conventional sliding mode control process, therefore,it has been customary to construct a single hyperplane in a manner tokeep in balance the quick response or stability of convergence of statequantities toward a balanced point on the hyperplane and the quickresponse or stability of convergence of the state quantities onto thehyperplane.

However, the sliding mode control process with the hyperplane thusconstructed is unable to simultaneously increase both the quick responseor stability of convergence of the state quantities toward the balancedpoint on the hyperplane and the quick response or stability ofconvergence of the state quantities onto the hyperplane. Consequently,it has been difficult to further increase the quick response orstability of convergence of state quantities of an object to becontrolled toward target state quantities.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide a slidingmode control method for controlling state quantities of an object to becontrolled according to a sliding mode control process, the sliding modecontrol method being capable of converging the state quantities onto ahyperplane and also toward a balanced point on the hyperplane withrespective degrees of quick response or stability of convergence.

Another object of the present invention is to provide a sliding modecontrol method which is capable of increasing both the quick response orstability of convergence of state quantities onto a hyperplane and thequick response or stability of convergence of the state quantitiestoward a balanced point on the hyperplane, for thereby furtherincreasing the quick response or stability of convergence of the statequantities toward target state quantities.

To accomplish the above objects, there is provided in accordance withthe present invention a sliding mode control method comprising the stepsof establishing a hyperplane for a sliding mode control process with alinear function having as variables a plurality of state quantities ofan object to be controlled, converging the state quantities onto thehyperplane, converging the state quantities toward a balanced point onthe hyperplane while converging the state quantities onto thehyperplane, thereby to control the state quantities at target statequantities represented by the balanced point, and variably establishingthe hyperplane depending on the manner in which the state quantities areconverged onto the hyperplane.

Since the hyperplane is variably established depending on the manner inwhich the state quantities are converged onto the hyperplane, the statequantities are converged onto the hyperplane and also converged towardthe balanced point on the hyperplane with respective degrees of quickresponse or stability of convergence of the state quantities.Consequently, the state quantities of the object to be controlled can becontrolled at the target state quantities with a desired quick responseor stability of convergence.

It is determined whether the state quantities are substantiallyconverged onto the hyperplane. If the state quantities are substantiallyconverged onto the hyperplane, the hyperplane is gradually varied inorder to increase the stability of convergence of the state quantitiestoward the balanced point or shorten the time (increase the quickresponse) required for the state quantities to converge toward thebalanced point.

Consequently, it is possible to increase the stability or quick responseof convergence of the state quantities toward the balanced point on thehyperplane, in addition to the intrinsic high stability of the slidingmode control process.

If the state quantities are not substantially converged onto thehyperplane, the hyperplane is established in order to substantiallyminimize the time required for the state quantities to converge onto thehyperplane.

Thus, the time required for the state quantities to converge onto thehyperplane is reduced (the quick response for convergence is shortened),resulting in a reduction in the time required for the state quantitiesto converge toward the balanced point on the hyperplane.

Accordingly, it is possible to increase both the stability or quickresponse of convergence of the state quantities onto the hyperplane andthe stability or quick response of convergence of the state quantitiestoward the balanced point on the hyperplane, for thereby furtherincreasing the stability or quick response of convergence of the statequantities toward the target state quantities.

On the hyperplane, the linear function has a value of "0". Therefore, itis possible to determine whether the state quantities are substantiallyconverged onto the hyperplane or not by comparing the value of thelinear function with a predetermined value.

If the state quantities comprise two state quantities (at this time, thehyperplane is straight), then the gradient of the hyperplane isgradually increased when the state quantities are substantiallyconverged onto the hyperplane, for thereby increasing the stability andquick response of convergence of the state quantities toward thebalanced point.

Specifically, a predetermined positive numerical value and apredetermined negative numerical value are established in eachpredetermined cycle time depending on whether the state quantities aresubstantially converged onto the hyperplane or not, the establishednumerical values are integrated, and the hyperplane is established byvarying the gradient of the hyperplane from a predetermined initialvalue of the gradient by an amount corresponding to the integratedvalue.

The established positive and negative numerical values are integrateddepending on whether the state quantities are substantially convergedonto the hyperplane or not When the state quantities are moving towardand away from the hyperplane and hence are not sufficiently convergedonto the hyperplane, the integrated numerical value is not largelychanged. When the state quantities are sufficiently converged onto thehyperplane, the numerical value is substantially steadily maintained asa positive numerical value. Therefore, the integrated value increases,gradually increasing the gradient of the hyperplane, resulting in anincrease in the stability of convergence of the state quantities towardthe balanced point and a reduction in the time required for the statequantities to converge toward the balanced point.

The initial value of the gradient is determined to substantiallyminimize the time required for the state quantities to converge onto thehyperplane if the state quantities are not substantially converged ontothe hyperplane. Thus, the state quantities can be converged onto thehyperplane within a short time.

It is preferable to vary the gradient of the hyperplane within apredetermined range. With limits thus provided for the gradient of thehyperplane, the gradient of the hyperplane is prevented from becomingtoo large or too small.

Preferably, the state quantities are converged onto the hyperplaneaccording to an adaptive sliding control mode including reaching andadaptive control laws. The adaptive sliding control mode serves toconverge the state quantities onto the hyperplane in view of effects ofdisturbances, etc. using the adaptive control law in addition to thereaching control law of the ordinary sliding mode control process.According to the adaptive sliding control mode, the state quantities canbe converged toward the target state quantities with high stabilityagainst disturbances, etc.

The reaching control law is a control law for converging the statequantities onto the hyperplane when there is no effect of disturbancesin the adaptive sliding mode control process. The adaptive control lawis a control law for compensating for an effect of disturbances when thestate quantities are converged onto the hyperplane in the adaptivesliding mode control process.

The above and other objects, features, and advantages of the presentinvention will become apparent from the following description when takenin conjunction with the accompanying drawings which illustrate preferredembodiments of the present invention by way of example.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an air-fuel control system for an internalcombustion engine according to an embodiment of the present invention,to which a sliding mode control method according to the presentinvention is applied;

FIG. 2 is a diagram showing output characteristics of an O₂ sensor usedin the air-fuel control system shown in FIG. 1;

FIG. 3 is a diagram illustrative of a model of an object to becontrolled in the air-fuel control system shown in FIG. 1;

FIG. 4 is a block diagram of the model shown in FIG. 3;

FIG. 5 is a block diagram of a model used as an estimator in a statepredictor in the air-fuel control system shown in FIG. 1;

FIG. 6 is a block diagram of the state predictor in the air-fuel controlsystem shown in FIG. 1;

FIG. 7 is a diagram illustrative of a sliding mode control process;

FIG. 8 is a diagram illustrative of the position of a pole in a controlunit in the air-fuel control system shown in FIG. 1;

FIG. 9 is a block diagram of an adaptive sliding mode controller in theair-fuel control system shown in FIG. 1,

FIG. 10 is a diagram illustrative of a hyperplane used by the adaptivesliding mode controller shown in FIG. 9;

FIG. 11 is a block diagram of an adaptive controller in the air-fuelcontrol system shown in FIG. 1;

FIG. 12 is a diagram illustrative of the timing to calculate an outputfuel injection quantity and a basic air-fuel ratio correction quantityin the air-fuel control system shown in FIG. 1;

FIG. 13 is a flowchart of an operation sequence of the air-fuel controlsystem shown in FIG. 1;

FIG. 14 is a flowchart of an operation sequence of the air-fuel controlsystem shown in FIG. 1;

FIG. 15 is a flowchart of an operation sequence of the air-fuel controlsystem shown in FIG. 1;

FIG. 16 is a flowchart of an operation sequence of the air-fuel controlsystem shown in FIG. 1;

FIGS. 17(a) through 17(c) are diagrams showing the results of asimulation process effected on the air-fuel control system shown in FIG.1 and a conventional air-fuel control system; and

FIG. 18 is a block diagram of an air-fuel control system for an internalcombustion engine according to another embodiment of the presentinvention, to which the sliding mode control method according to thepresent invention is applied.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A sliding mode control method according to the present invention will bedescribed as a process of controlling the air-fuel ratio of an internalcombustion engine.

FIG. 1 shows in block form an air-fuel control system for an internalcombustion engine according to an embodiment of the present invention.As shown in FIG. 1, an internal combustion engine 1 such as afour-cylinder internal combustion engine on which air-fuel ratio controlis to be effected includes exhaust pipes 2 extending from the respectivecylinders and joined together to a single main exhaust pipe 3 near thecylinder block. Two three-way catalytic converters 4, 5 are mounted inthe main exhaust pipe 3 at successively downstream locations thereon.The downstream catalytic converter 5 may be dispensed with.

The air-fuel control system combined with the internal combustion engine1 comprises a wide-range air-fuel ratio sensor 6 mounted as a firstexhaust gas sensor on the junction of the exhaust pipes 2 upstream ofthe catalytic converter 4, an O₂ sensor (oxygen concentration sensor) 7mounted as a second exhaust gas sensor on the main exhaust pipe 3downstream of the catalytic converter 4, and a control unit 8 forcarrying out a control process (described later on) based on detectedoutput signals from the sensors 6, 7. The control unit 8 is suppliedwith detected output signals from the sensors 6, 7 and also detectedoutput signals from various other sensors including a engine speedsensor, an intake pressure sensor, a coolant temperature sensor, etc.

The wide-range air-fuel ratio sensor 6 is in the form of an O₂ sensor,and outputs a signal having a level depending on the concentration ofoxygen (which is commensurate with the air-fuel ratio of an air-fuelmixture that is supplied to the internal combustion engine 1)representative of the air-fuel ratio of an exhaust gas in the junctionof the exhaust pipes 2 upstream of the catalytic converter 4. The outputsignal from the wide-range air-fuel ratio sensor 6 passes through afilter 9 in the control unit 8 which removes high-frequency noises fromthe output signal, and then is converted by a linearizer 10 in thecontrol unit 8 into a signal having a level which is proportional to theoxygen concentration (air-fuel ratio) of an exhaust gas in a wide rangeof oxygen concentrations. The wide-range air-fuel ratio sensor 6 whoseoutput signal will thus be linearized will hereinafter be referred to asan LAF sensor 6.

The O₂ sensor 7 disposed downstream of the catalytic converter 4 outputsa signal having a level depending on the oxygen concentration (which iscommensurate with the air-fuel ratio of the exhaust gas that has passedthrough the catalytic converter 4) of the exhaust gas that has passedthrough the catalytic converter 4. As shown in FIG. 2, the output signalfrom the O₂ sensor 7 is substantially proportional with high sensitivityto the oxygen concentration of the exhaust gas that has passed throughthe catalytic converter 4, with the air-fuel ratio of the air-fuelmixture supplied to the internal combustion engine 1 (the air-fuel ratioof the exhaust gas emitted from the internal combustion engine 1) beingin a range close to a predetermined appropriate value. High-frequencynoises are removed from the output signal of the O₂ sensor 7 by thefilter 11 in the control unit 8.

The control unit 8 comprises a microcomputer and has as its mainfunctions a basic fuel injection quantity calculator 12 for determininga basic fuel injection quantity Tim to be injected into the internalcombustion engine 1, a first correction coefficient calculator 13 fordetermining a first correction coefficient KTOTAL to correct the basicfuel injection quantity Tim in view of an exhaust recirculation ratio(the proportion of the exhaust gas contained in intake air of theinternal combustion engine 1) of the internal combustion engine 1, apurged quantity of fuel supplied to the internal combustion engine 1when a canister (not shown) thereof is purged, the coolant temperatureand intake temperature of the internal combustion engine 1, etc., asecond correction coefficient calculator 14 for determining a secondcorrection coefficient KCMDM to correct the basic fuel injectionquantity Tim in view of the charging efficiency of intake aircorresponding to a target air-fuel ratio from the target air-fuel ratioat the LAF sensor 6, a basic air-fuel ratio setting unit 15 forestablishing a basic air-fuel ratio KBS (a basic air-fuel ratio at theLAF sensor 6) of the internal combustion engine 1, a target air-fuelratio calculator 16 for correcting the basic air-fuel ratio KBS based onthe output signal from the O₂ sensor 7 thereby to determine a targetair-fuel ratio KCMD at the LAF sensor 6, and a feedback controller 17for feedback-controlling a fuel injection quantity (fuel supplyquantity) of the internal combustion engine 1 based on the output signalfrom the LAF sensor 6 so as to converge the air-fuel ratio at the LAFsensor 6 toward the target air-fuel ratio KCMD.

The basic fuel injection quantity calculator 12 determines a referencefuel injection quantity from the rotational speed and intake pressure ofthe internal combustion engine 1 using a predetermined map, and correctsthe determined reference fuel injection quantity depending on theeffective opening area of a throttle valve (not shown) of the internalcombustion engine 1, thereby calculating a basic fuel injection quantityTim.

Specific methods of calculating the basic fuel injection quantity Tim,the first correction coefficient KTOTAL, and the second correctioncoefficient KCMDM are disclosed in Japanese laid-open patent publicationNo. 5-79374 which corresponds to U.S. Pat. No. 5,253,630, and will notbe described in detail below. The basic fuel injection quantity Tim iscorrected by being multiplied by the first correction coefficient KTOTALand the second correction coefficient KCMDM, producing a demand fuelinjection quantity Tcyl.

The basic air-fuel ratio setting unit 15 determines a basic air-fuelratio KBS from the rotational speed and intake pressure (whichrepresents the load on the internal combustion engine 1) of the internalcombustion engine 1 using a predetermined map.

The target air-fuel ratio calculator 16 comprises a state predictor 18for estimating state quantities (specifically, the oxygen concentrationat the O₂ sensor 7 and a changing degree such as a change or rate ofchange of the oxygen concentration at the O₂ sens or 7) in an exhaustsystem A which extends from the LAF sensor 6 to the O₂ sensor 7 andincludes the catalytic converter 4, in view of a dead time present inthe exhaust system A, and an adaptive sliding mode controller 19(correction quantity calculating means) for determining a correctionquantity for the basic air-fuel ratio KBS based on the state quantitiesestimated by the state predictor 18 according to an adaptive slidingmode control process. The target air-fuel ratio calculator 16 calculatesthe target air-fuel ratio KCMD by correcting the basic air-fuel ratioKBS with the determined correction quantity, i.e., adding the correctionquantity to the basic air-fuel ratio KBS. Details of the state predictor18 and the adaptive sliding mode controller 19 will be described lateron.

The feedback controller 17 comprises a general feedback controller 20for feedback-controlling a total fuel injection quantity for all thecylinders of the internal combustion engine 1 so as to converge theair-fuel ratio detected by the LAF sensor 6 toward the target air-fuelratio, and a local feedback controller 21 for feedback-controlling atotal fuel injection quantity for each of the cylinders of the internalcombustion engine 1.

The general feedback controller 20 determines a feedback correctioncoefficient KFB to correct the demand fuel injection quantity Tcyl so asto converge the air-fuel ratio detected by the LAF sensor 6 toward thetarget air-fuel ratio. The general feedback controller 20 comprises aPID controller 22 for determining a feedback correction coefficient KFBfrom the detected air-fuel ratio from the LAF sensor 6 and the targetair-fuel ratio according to a known PID control process so as toeliminate any difference between the detected air-fuel ratio from theLAF sensor 6 and the target air-fuel ratio, and an adaptive controller23 (indicated by "STR" in FIG. 1) which is a recursive-type controllerfor adaptively determining a feedback correction coefficient KFB fromthe detected air-fuel ratio from the LAF sensor 6 and the targetair-fuel ratio in view of dynamic changes such as changes in operatingconditions of the internal combustion engine 1 or characteristic changesthereof. The feedback correction coefficients KFB separately determinedby the PID controller 22 and the adaptive controller 23 are selected oneat a time by a switcher 24, and the demand fuel injection quantity Tcylis corrected by being multiplied by the selected feedback correctioncoefficient KFB. The feedback correction coefficient KFB determined bythe PID controller 22 will hereinafter be referred to as "a feedbackcorrection coefficient KLAF" and the feedback correction coefficient KFBdetermined by the adaptive controller 23 will hereinafter be referred toas "a feedback correction coefficient KSTR". Details of the generalfeedback controller 20 will be described later on.

The output signal from the LAF sensor 6 is supplied to the PIDcontroller 22 and the adaptive controller 23 through respective filters24, 25 having respective frequency bands that match the respectivecontrol characteristics of the PID controller 22 and the adaptivecontroller 23.

The local feedback controller 21 comprises an observer 26 for estimatinga real air-fuel ratio #nA/F (n=1, 2, 3, 4) of each of the cylinders fromthe air-fuel ratio detected by the LAF sensor 6 (the air-fuel ratio inthe junction of the exhaust pipes 2 extending from the respectivecylinders of the internal combustion engine 1), and a plurality of PIDcontrollers 27 (as many as the number of the cylinders) for determininga feedback correction coefficient #nKLAF for a fuel injection quantityfor each of the cylinders from the real air-fuel ratio #nA/F of each ofthe cylinders according to a PID control process so as to eliminatevariations of the air-fuel ratios of the cylinders.

Briefly stated, the observer 26 estimates a real air-fuel ratio #nA/F ofeach of the cylinders as follows: A system from the internal combustionengine 1 to the LAF sensor 6 is considered to be a system for beingsupplied with a real air-fuel ratio #nA/F of each of the cylinders andoutputting an air-fuel ratio detected by the LAF sensor 6 to thejunction of the exhaust pipes 2, and is modeled in view of a detectionresponse delay (e.g., a time lag of first order) of the LAF sensor 6 anda chronological contribution of the air-fuel ratio of each of thecylinders to the air-fuel ratio in the junction of the exhaust pipes 2.Based on the modeled system, a real air-fuel ratio #nA/F of each of thecylinders is estimated from the air-fuel ratio detected by the LAFsensor 6.

Details of the observer 26 are disclosed in Japanese laid-open patentpublication No. 7-83094 which corresponds to U.S. Pat. No. 5,531,208,for example, and will not be described below.

Each of the PID controllers 27 of the local feedback controller 21divides the air-fuel ratio detected by the LAF sensor 6 by an averagevalue of the feedback correction coefficients #nKLAF determined by therespective PID controllers 27 in a preceding cycle time to produce aquotient value, and uses the quotient value as a target air-fuel ratiofor the corresponding cylinder. Each of the PID controllers 27 thendetermines a feedback correction coefficient #nKLAF in a present cycletime so as to eliminate any difference between the target air-fuel ratioand the corresponding real air-fuel ratio #nA/F determined by theobserver 26. The local feedback controller 21 multiplies a value, whichhas been produced by multiplying the demand fuel injection quantity Tcylby the selected feedback correction coefficient KFB produced by thegeneral feedback controller 20, by the feedback correction coefficient#nKLAF for each of the cylinders, thereby determining an output fuelinjection quantity #nTout (n=1, 2, 3, 4) for each of the cylinders.

The output fuel injection quantity #nTout (n=1, 2, 3, 4) thus determinedfor each of the cylinders is corrected for accumulated fuel particles onintake pipe walls of the internal combustion engine 1 by a fuelaccumulation corrector 28 in the control unit 8. The corrected outputfuel injection quantity #nTout is applied to each of fuel injectors (notshown) of the internal combustion engine 1, which injects fuel into eachof the cylinders with the corrected output fuel injection quantity#nTout. The correction of the output fuel injection quantity in view ofaccumulated fuel particles on intake pipe walls is disclosed in detailin Japanese laid-open patent publication No. 8-21273 which correspondsto U.S. Pat. No. 5,568,799, for example, and will not be described indetail below.

Details of the state predictor 18 and the adaptive sliding modecontroller 19 of the target air-fuel ratio calculator 16 will bedescribed below.

The target air-fuel ratio calculator 16 serves to correct the basicair-fuel ratio KBS into a target air-fuel ratio KCMD at the LAF sensor 6upstream of the catalytic converter 4 so as to adjust the oxygenconcentration of the exhaust gas at the O₂ sensor downstream of thecatalytic converter 4 to a predetermined adequate value which maximizesthe purifying capability of the catalytic converter 4. The targetair-fuel ratio calculator 16 has an object (a plant) to be controlledwhich comprises the exhaust system A which extends from the LAF sensor 6to the O₂ sensor 7 and includes the catalytic converter 4. A value withwhich to correct the basic air-fuel ratio KBS is determined by the statepredictor 18 and the adaptive sliding mode controller 19 according tothe adaptive sliding mode control process in view of the dead timepresent in the exhaust system A. The air-fuel ratio at the LAF sensor 6will hereinafter be referred to as "pre-CAT A/F", and the oxygenconcentration at the O₂ sensor 7 will hereinafter be referred to as"post-CAT A/F".

To make the adaptive sliding mode control process applicable in view ofa dead time present in the exhaust system A which is an object to becontrolled (hereinafter referred to as an "object exhaust system A"),the object exhaust system A is modeled by a spring mass system (with atime lag of second order) including a dead time, as shown in FIG. 3.

In spring mass system shown in FIG. 3, a mass body 29 (whose mass M isassumed to be "1") is supported by a spring 30 having a spring constantK and a damper 31 having a damping coefficient C. A vibrating forceapplied to the mass body 29 corresponds to the pre-CAT A/F, and adisplacement x₁ of the mass body 29 which is caused by the vibratingforce corresponds to the post-CAT A/F. The pre-CAT A/F is the sum of anair-fuel ratio component u (referred to as an "input u") controllable bythe feedback controller 17, etc., and an air-fuel ratio component L(referred to as a "disturbance L") such as noises that are notcontrollable by the feedback controller 17. The input u and thedisturbance L contain a dead time d in the exhaust system A. The input u(t--d) and the disturbance L (t--d) prior to the dead time d are appliedas a vibrating force to the spring mass system.

If it is assumed in the model of the spring mass system that a value ofthe post-CAT A/F which corresponds to a displacement of the mass body 29is represented by x₁ and a rate of change thereof by x₂, then using thespring constant K, the damping coefficient C, etc., state equations ofthe model are given as follows: ##EQU1##

The above state equations (1) are expressed in the block diagram of FIG.4, which shows a plant model of the object exhaust system A. In FIG. 4,the letter "s" indicates a Laplace operator.

The state predictor 18 and the adaptive sliding mode controller 19 areconstructed on the basis of the plant model of the object exhaust systemA, and will be described in detail below.

The state predictor 18 serves to compensate for the dead time d in theobject exhaust system A in an adaptive sliding mode control process thatis carried out by the adaptive sliding mode controller 19. The statepredictor 18 estimates a state quantity of the post-CAT A/F which isdetected by the O₂ sensor 7 after the dead time d in the object exhaustsystem A so as to correspond to the pre-CAT A/F up to the present time,from the pre-CAT A/F detected by the LAF sensor 6 and the post-CAT A/Fdetected by the O₂ sensor 7. In this embodiment, the state quantitycomprises two values, i.e., the value of the post-CAT A/F detected bythe O₂ sensor 7 (actually the output level of the O₂ sensor 7) and achange or rate of change (actually a change or rate of change of theoutput level of the O₂ sensor 7) of the value of the post-CAT A/F.

In order to estimate the state quantity, the state predictor 18 isconstructed to effect the following processing:

The state predictor 18 uses for the estimation a model (plant model) ofa delay element shown in FIG. 5 which is similar to the plant modelshown in FIG. 5 except that the term of the dead time (expressed by"e^(-ds) " in FIG. 4) is dispensed with and the constants C, K, b arereplaced with preset values C_(M), K_(M), b_(M), respectively. In themodel of the delay element shown in FIG. 5, state equations, whichcorresponds to the state equations (1), are given as follows: ##EQU2##where U=u+L.

In FIG. 6 and the equations (2), x_(1M), x_(2M) represent the value ofthe post-CAT A/F and a change or rate of change thereof (state quantity)in the model of the delay element shown in FIG. 5. The preset valuesC_(M), K_(M), b_(M) are determined by experimentation or the like.

The state predictor 18 uses, as the input U(t) in the equations (2), thepre-CAT A/F actually detected by the LAF sensor 6, and solves theequations (2) in a time-series for state quantities x_(1M), x_(2M).Furthermore, the state predictor 18 determines estimated values x₁ hat(=an estimated value of the post-CAT A/F), x₂ hat (=an estimated valueof a change or rate of change of the post-CAT A/F) of the post-CAT A/Fafter the dead time d from the present time t from the determined statequantities x_(1M), X_(2M) and state quantities x₁, x₂ of the post-CATA/F at the present time t, according to the following equation (3):##EQU3## where "e^(At) " represents a matrix exponential functionobtained when the state equations (2) are solved, and "d_(M) " a presetvalue (identified value) of the dead time d in the object exhaust systemA. The dead time d_(M) is equal to the actual dead time d or set to agreater value (d_(M) ≧d). For the first term of the equation (3), thestate quantities x₁, x₂ (the value of the post-CAT A/F and a change orrate of change thereof) which are actually obtained from the outputsignal from the O₂ sensor are used.

In the equation (3), the first term of the right side is a term forestimating a state quantity of the post-CAT A/F detected by the O₂sensor after the dead time d if the input U (the pre-CAT A/F from thetime t-d to the time t) which is applied to the object exhaust system Afrom the present time t to the time t+d after the dead time d in theobject exhaust system A is "0".

The second and third terms of the right side are terms for estimating achange in the state quantity of the post-CAT A/F detected by the O₂sensor after the dead time d with the input U (the pre-CAT A/F from thetime t-d to the time t) which is applied to the object exhaust system Afrom the present time t to the time t+d after the dead time d in theobject exhaust system A.

The state predictor 18 which effects the above estimating operations isshown in block form in FIG. 6. As shown in FIG. 6, the state predictor18 generally comprises an estimator 32 for carrying out the estimatingoperation represented by the first term of the right side of theequation (3) and an estimator 33 for carrying out the operation to solvethe state equations (2) and the estimating operation represented by thesecond and third terms of the right side of the equation (3).

The estimator 32 is given the state quantities (the value x₁ of thepost-CAT A/F and a change or rate of change x₂ thereof) actuallyobtained from the output signal from the O₂ sensor 7. The statequantities obtained from the output signal from the O₂ sensor 7 arefiltered or scaled, if necessary, by an element 34, and then applied tothe estimator 32. In FIG. 6, the value x₁ of the post-CAT A/F and achange or rate of change x₂ thereof are shown as being supplied directlyfrom the O₂ sensor 7 to the estimator 32 for illustrative purposes.Actually, however, a change or rate of change x₂ of the post-CAT A/F iscalculated in the control unit 8.

To effect the above estimating operations, the estimator 33 is given thepre-CAT A/F actually obtained from the output signal from the LAF sensor6 as the input U (=u+L). The pre-CAT A/F obtained from the output signalfrom the LAF sensor 6 is filtered or scaled, if necessary, by an element35, and then applied to the estimator 32.

The state predictor 18 adds values determined by the respectiveestimators 32, 33, and outputs the sum as estimated values x₁ hat, x₂hat of the state quantities of a the post-CAT A/F detected by the O₂sensor 7 after the dead time d, to the adaptive sliding mode controller19. The values determined by the respective estimators 32, 33 are addedafter being filtered and scaled, if necessary, by respective elements36, 37. The sum (the estimated values x₁ hat, x₂ hat of the statequantities of the post-CAT A/F after the dead time d) is also filteredand scaled, if necessary, by an element 38, and then outputted to theadaptive sliding mode controller 19. The estimated values x₁ hat, x₂ hatwill hereinafter be referred to as estimated state quantities x₁ hat, x₂hat, respectively.

The adaptive sliding mode controller 19 will be described in detailbelow.

A general sliding mode control process will first briefly be describedbelow with reference to FIG. 7.

The sliding mode control process is a feedback control process ofvariable structure. According to the sliding mode control process, ifthere two state quantities x₁, x₂ of an object to be controlled, then ahyperplane H expressed by σ=0 is designed beforehand using a linearfunction σ=s₁ x₁ +s₂ x₂ (s₁, s₂ are coefficients) with the statequantities x₁, x₂ used as variables therein. The linear function σ iscalled a switching function. If the degree of a phase plane is larger,then a switching line changes to a switching plane and then to ahyperplane which cannot geometrically be illustrated. The hyperplane mayalso be called a slip plane.

When the state quantities x₁, x₂ are such that σ≠0 as indicated by apoint P in FIG. 7, the state quantities x₁, x₂ are caused to converge ata high speed onto the hyperplane H (σ=0) under high gain controlaccording to the so-called reaching control law (mode 1), and then toconverge toward a balanced point (a converged point which is a pointwhere x₁ =x₂ =0) on the hyperplane H while converging onto thehyperplane H according to the so-called equivalent control input (mode2).

In the sliding mode control process, the state quantities x₁, x₂ canconverge highly stably toward the balanced point on the hyperplane Haccording to the equivalent control input without being affected by adisturbance, etc. simply when the state quantities x₁, x₂ are convergedonto the hyperplane H. Therefore, it is important to stably converge thestate quantities x₁, x₂ onto the hyperplane H in the mode 1. If there isa disturbance, then it is generally difficult to converge the statequantities x₁, x₂ stably onto the hyperplane H according to only thereaching control law. In view of this, there has in recent years beenproposed an adaptive sliding mode control process which employs anadaptive control law for converging state quantities onto a hyperplanewhile eliminating the effect of a disturbance, in addition to thereaching control law, as disclosed in, for example, "Sliding modecontrol--design theory of nonlinear robust control--", pages 134˜135,published Oct. 20, 1994 by Corona Co., Ltd.

The adaptive sliding mode controller 19 uses such an adaptive slidingmode control process to calculate a correction quantity for the basicair-fuel ratio from the estimated state quantities x₁ hat, x₂ hat of thepost-CAT A/F. The adaptive sliding mode controller 19 is constructed asfollows:

First, the construction of a hyperplane required for the adaptivesliding mode control process of the adaptive sliding mode controller 19and the equivalent control input will first be described below.

In this embodiment, the adaptive sliding mode controller 19 determines acorrection quantity for the basic air-fuel ratio KBS in order to adjustthe post-CAT A/F to a predetermined adequate value. Therefore, targetvalues for the estimated state quantities x₁ hat, x₂ hat of the post-CATA/F (the estimated value of the post-CAT A/F after the dead time d andthe estimated value of its change or rate of change), i.e., valuestoward which the estimated state quantities x₁ hat, x₂ hat of thepost-CAT A/F are to converge, are set to "an appropriate value" and "0",respectively.

A hyperplane for carrying out the adaptive sliding mode control processwith the appropriate value of the post-CAT A/F being "q" is expressed bya linear function according to the following equation (4):

    σ=S.sub.1 ·(x.sub.1 -q)+S.sub.2 ·x.sub.2 =0 (4)

If the estimated state quantities x₁ hat, x₂ hat are used, then sincethe dead time d is compensated for by the state predictor 18, the plantmodel of the object exhaust system A is represented by the structureshown in FIG. 5 where the state quantities x_(1M), x_(2M) are replacedwith the estimated state quantities x₁ hat, x₂ hat.

Therefore, state equations of the plant model are expressed as follows:##EQU4##

By effecting a linear transformation expressed by the followingequations (6) based on the equation (4) in the state equations (5):##EQU5## and by replacing the disturbance L with "0", the followingequations (7) are obtained: ##EQU6##

If the sliding mode control process is carried out using the hyperplaneexpressed by the equation (4), then in the mode 2 for converging theestimated state quantities x₁ hat, x₂ hat toward the balanced point onthe hyperplane while converging them onto the hyperplane, it isnecessary to satisfy the following equations: ##EQU7##

Therefore, from the equations (7), an equivalent control input u_(eq)(=u) required in the mode 2 is indicated by the following equation (9):##EQU8##

Then, when the estimated state quantities x₁ hat, x₂ hat are convergedonto the hyperplane by the equivalent control input u_(eq), since σ=0,the following equation (10) is obtained from the lower one of theequations (7): ##EQU9##

For the sake of brevity, s₁ =k, s₂ =1 (k=s₁ /s₂). In view of the factthat the target value q (the appropriate value of the post-CAT A/F) forthe estimated quantity x₁ hat is "0" (constant value) when the time t<0and "q" (constant value) when the time t≧0, the equation (10) isLaplace-transformed into the following equation (11): ##EQU10## where x₁hat represents a Laplace transformation of the estimated state quantityx₁ hat, and s represents a Laplace operator.

Consequently, when the equation (11) is inversely Laplace-transformed,the estimated state quantity x₁ hat is expressed on the time base by thefollowing equation (12):

    x.sub.1 =-q·e.sup.-k.t +q                         (12)

If k>0(s₁ >0, s₂ =1) in the equation (12), then the estimated statequantity x₁ hat converges toward the target value q at t→∞. This meansthat the characteristic root of the equation (11)-k (the pole of thecontrol system) is positioned in a stable region on a complex plane (aregion where the real part of the pole is negative), as shown in FIG. 8.

Therefore, the hyperplane used in this embodiment is establishedaccording to the following equation (13):

    σ=k(x.sub.1 -q)+x.sub.2 =0(k>0)                      (13)

A specific value of k in the equation (13) is established based onvarious experiments and simulations basically such that the estimatedstate quantities x₁ hat, x₂ hat will quickly converge onto thehyperplane. In this embodiment, the value of k may be varied as desiredas described later on. In this embodiment, since the adaptive slidingmode controller 19 is constructed as a servo-type controller, the targetvalue q for the estimated state quantity x₁ hat is q≠0. However, theadaptive sliding mode controller 19 may be constructed as aregulator-type controller in the same manner as with the aboveembodiment with the target value q for the estimated state quantity x₁hat being q=0.

The reaching control law of the sliding mode control process is acontrol law for converging the linear function σ onto the hyperplane(σ=0). Various types of the reaching control law are known in the art.In this embodiment, the acceleration rate law which is characterized bythe shortest time required for convergence onto the hyperplane amongthose various types of the reaching control law is employed.

According to the acceleration rate law, a dynamic characteristic of σ (arate of change of the value of σ over time) is controlled so that can beexpressed by the following equation (14):

    σ=-J·|σ|.sup.α ·sgn(σ)                                    (14)

where J, α are preset positive constants with 0<α<1, and sgn(σ) is asignum function of σ, sgn(σ)=-1 when σ<0, sgn(σ)=0 when σ=0, andsgn(σ)=1 when σ>0.

When the equation (13) is differentiated with respect to time, and thestate equations (5) are used with the disturbance L=0, the followingequation (15) is obtained:

    σ=(k-C.sub.M)·x.sub.2 K.sub.M ·x.sub.1 +b.sub.M ·u                                               (15)

From the equations (14) and (15), an input u_(s1) (=u) to the objectexhaust system A is represented by the following equation (16):##EQU11##

The input u_(s1) to the object exhaust system A, which is expressed bythe equation (16), is an input (the pre-CAT A/F) to be given to theobject exhaust system A in order to adjust the post-CAT A/F to theadequate value q when the disturbance L=0. The first and second terms ofthe equation (16) agree with the equivalent control input u_(eq)expressed by the equation (9) when σ=0, i.e., when the estimated statequantities x₁ hat, x₂ hat are converged onto the hyperplane, asdescribed below. ##EQU12##

This is made apparent by establishing s₁ /s₂ =k in the equation (9),expressing the value q with x₁ hat, x₂ hat using the equation (13), andsubstituting it in the equation (9).

The third term of the equation (13) represents a control input forconverging the estimated state quantities x₁ hat, x₂ hat onto thehyperplane according to the reaching control law when the disturbanceL=0. A control input according to the reaching control law willhereinafter be referred to as a reaching control input u_(rch) which isexpressed by: ##EQU13##

The adaptive control law of the adaptive sliding mode control process inthe adaptive sliding mode controller 19 is constructed as follows:

As described above, the hyperplane, the equivalent control input u_(eq),and the reaching control input u_(rch) are constructed on the assumptionthat the disturbance L=0. Actually, however, various disturbances existin the object exhaust system A, and the plant model used in constructingthe hyperplane, etc. suffers a model error with respect to the actualobject exhaust system A. If the estimated state quantities x₁ hat, x₂hat are converged onto the hyperplane, then the estimated statequantities x₁ hat, x₂ hat are converged toward the balanced point on thehyperplane without being affected by the disturbances and the modelerror. At a stage in which the estimated state quantities x₁ hat, x₂ hatare not converged onto the hyperplane, the estimated state quantities x₁hat, x₂ hat are cannot be converged onto the hyperplane with thereaching control input u_(rch) according to the reaching control law.

The adaptive control law used in the adaptive sliding mode controller 19serves to eliminate the above drawback.

According to the present embodiment, for constructing the adaptivecontrol law in the adaptive sliding mode controller 19, it is assumedthat the disturbance L is invariable without depending on time and theestimated state quantities x₁ hat, x₂ hat, and an integrated termu_(adp) of the linear function σ which is represented by the followingequation (19) is added as an adaptive control law term ("u_(adp) " willhereinafter be called an "adaptive control input") to the right side ofthe equation (16), determining a final input u_(s1) to the objectexhaust system A. ##EQU14##

Therefore, the input u_(s1) to the object exhaust system A using theadaptive control law is determined according to the following equation(20): ##EQU15##

The equation (20) represents the simplest form of the adaptive slidingmode control process. It is also possible to employ a more developedform of adaptive control law.

The adaptive sliding mode controller 19 effects calculations accordingto the equation (20) to determine an input u_(s1) to the object exhaustsystem A. According to the present embodiment, since the basic air-fuelratio KBS is corrected so as to adjust the estimated state quantity x₁hat to the appropriate value q (x₁ hat=q, x₂ hat=0) for therebyindirectly adjusting the post-CAT A/F to the appropriate value q, theadaptive sliding mode controller 19 outputs the input u_(s1) determinedby the equation (20) as a correction quantity for the basic air-fuelratio KBS. The input u_(s1) determined by the equation (20) willhereinafter be referred to as a "basic air-fuel ratio correctionquantity u_(s1) ".

The adaptive sliding mode controller 19 thus constructed as describedabove is shown in block form in FIG. 9. As shown in FIG. 9, the adaptivesliding mode controller 19 mainly comprises an equivalent control inputcalculator 39 for determining the equivalent control input u_(eq), and anonlinear input calculator 40 for determining the sum u_(n1) (=u_(rch)+u_(adp), hereinafter referred to as a "nonlinear input") of thereaching control input u_(rch) and the adaptive control input u_(adp).These calculators 39, 40 are supplied with the estimated statequantities x₁ hat, x₂ hat determined by the state predictor 18 throughthe element 38.

Basically, the adaptive sliding mode controller 19 outputs the basicair-fuel ratio correction quantity u_(s1), which is the sum of theequivalent control input u_(eq) determined by the equivalent controlinput calculator 39 and the nonlinear input u_(n1). The basic air-fuelratio correction quantity u_(s1) is scaled and filtered, if necessary,by an element 41, and then stored in a memory (not shown). The basicair-fuel ratio correction quantity u_(s1) is calculated in a cycle timehaving a predetermined constant period.

The target air-fuel ratio calculator 16, which has the adaptive slidingmode controller 19 and the state predictor 18, adds the basic air-fuelratio correction quantity u_(s1) stored in the memory to the basicair-fuel ratio KBS, thereby correcting the basic air-fuel ratio KBS intothe target air-fuel ratio KCMD. The target air-fuel ratio KCMD iscalculated by the target air-fuel ratio calculator 16 out of synchronismwith the calculation by the adaptive sliding mode controller 19 of thebasic air-fuel ratio correction quantity u_(s1), but in synchronism witha crankshaft angle period (so-called TDC) of the internal combustionengine 1, as described later on.

As shown in FIG. 9, the adaptive sliding mode controller 19 alsoincludes, in addition to the calculators 39, 40, a stability decisionunit 42 for determining stability of the adaptive sliding mode controlprocess, and a correction limiter 43 (correction quantity calculationlimiting means) for limiting correction of the basic air-fuel ratio KBSdepending on the determined stability of the adaptive sliding modecontrol process.

The stability decision unit 42 carries out a stability decision processshown in FIG. 16 each time the basic air-fuel ratio correction quantityu_(s1) is calculated. As shown in FIG. 16, the stability decision unit42 first determines a rate of change σ dot over time of the linearfunction σ (determined by the nonlinear input calculator 40 shown inFIG. 9) expressed by the equation (13), i.e., a differential of thelinear function σ with respect to time in STEP16-1. The stabilitydecision unit 42 then determines whether the absolute value of thelinear function σ is greater than a predetermined value σ₁ (|σ|>σ₁) orthe absolute value of the rate of change σ dot is greater than apredetermined value σ₂ (|σdot|>σ₂) in STEP16-2. If |σ|>σ₁ or |σdot|>σ₂(YES in STEP16-2), then the stability decision unit 42 decides that theadaptive sliding mode control process is unstable in STEP16-3, andfinishes the present decision cycle. A condition, thus determined asunstable, of the adaptive sliding mode control process is that theestimated state quantities x₁ hat, x₂ hat are largely spaced apart fromthe hyperplane (σ=0) or they are subjected to a large time-dependentchange in a direction away from the hyperplane (σ=0).

If the condition of STEP16-2 is not met (NO in STEP16-2), then thestability decision unit 42 determines whether the product σ·σ dot of thevalue of σ and the value of σ dot (which corresponds to a timedifferential function of the Lyapunov's function σ² /2 relative to σ) isgreater than a predetermined value a (≧0) (σ·σdot>a) in STEP16-4. Ifσ·σdot>a (YES in STEP16-4), then the stability decision unit 42 decidesthat the adaptive sliding mode control process is unstable in STEP16-3,and finishes the present decision cycle. If the condition of STEP16-4 isnot met (NO in STEP16-4), then the stability decision unit 42 decidesthat the adaptive sliding mode control process is stable in STEP16-5,and finishes the present decision cycle. A condition, thus determined asunstable, of the adaptive sliding mode control process is that theestimated state quantities x₁ hat, x₂ hat are shifted in a directionaway from the hyperplane (σ=0) on a side where σ² increases.

In this embodiment, the adaptive sliding mode control process isdetermined for stability according to the two conditions in STEP16-2,STEP16-4. However, the adaptive sliding mode control process may bedetermined for stability according to one of the two conditions inSTEP16-2, STEP16-4 or one of the two inequalities in STEP16-2.

Therefore, according to the stability decision process described above,the stability decision unit 42 decides that the adaptive sliding modecontrol process is unstable when the estimated state quantities x₁ hat,x₂ hat do not possibly converge onto the hyperplane (σ=0).

If the stability decision unit 42 decides that the adaptive sliding modecontrol process is unstable, then the correction limiter 43 (see FIG. 9)prevents the outputting of the basic air-fuel ratio correction quantityu_(s1) calculated by the adaptive sliding mode controller 19 in thepresent cycle time, and keeps the basic air-fuel ratio correct ionquantity u_(s1) calculated in the preceding cycle time as the output ofthe adaptive sliding mode controller 19, thereby limiting the correctionof the basic air-fuel ratio KBS by the basic air-fuel ratio correctionquantity u_(s1).

If the stability decision unit 42 decides that the adaptive sliding modecontrol process is stable, then the correction limiter 43 outputs thebasic air-fuel ratio correction quantity u_(s1) calculated by theadaptive sliding mode controller 19 in the present cycle time.

While the basic air-fuel ratio correction quantity u_(s1) calculated inthe preceding cycle time is kept as the output of the adaptive slidingmode controller 19 if the adaptive sliding mode control process isunstable in the illustrated embodiment, the basic air-fuel ratiocorrection quantity u_(s1) may forcibly be set to "0" thereby keepingthe basic air-fuel ratio KBS uncorrected if the adaptive sliding modecontrol process is unstable.

In the adaptive sliding mode control process according to the presentembodiment, when the estimated state quantities x₁ hat, x₂ hat convergeonto the hyperplane (σ=0) expressed by the equation (13) or a nearbyregion (σ≈0), the stability of convergence of the estimated statequantities x₁ hat, x₂ hat toward the target values "q", "0" (thebalanced point on the hyperplane) is higher as the gradient of thehyperplane (σ=0), stated otherwise, the value of the coefficient k (>0)in the equation (13), is greater. This is equivalent to the fact thatthe system stability is higher as the pole--k of the control systemshown in FIG. 8 becomes larger in a negative direction on the real axis.As can be seen from the equation (12), the greater the value of thecoefficient k, the shorter the time required for the estimated statequantities x₁ hat, x₂ hat to converge toward the target values "q", "0"on the hyperplane. From this standpoint, therefore, the coefficient kshould preferably be set to as large a value as possible.

If the value of the coefficient k in the equation (13) is too large,however, as long as the estimated state quantities x₁ hat, x₂ hat do notconverge onto the hyperplane (σ=0), the value of the linear function σis also large as can be understood from the equation (13), and hence thenonlinear input u_(n1) (=u_(rch) +u_(adp)) for converging the estimatedstate quantities x₁ hat, x₂ hat onto the hyperplane is also large. Ifthe nonlinear input u_(n1) is too large, then the estimated statequantities x₁ hat, x₂ hat tend to produce an oscillatory response withrespect to the hyperplane, resulting in an increase in the time requiredfor the estimated state quantities x₁ hat, x₂ hat to converge onto thehyperplane. Such an increase in the time reduces the stability ofconvergence and the quick response of the estimated state quantities x₁hat, x₂ hat. From this standpoint, therefore, it is not preferable toset the coefficient k to too a large value.

In view of the above considerations, the adaptive sliding modecontroller 19 according to this embodiment additionally has a hyperplanevariable controller 44 (hyperplane setting means) for varying the valueof the coefficient k in the equation (13) thereby to vary the hyperplaneof the adaptive sliding mode control process, as shown in FIG. 9.

The hyperplane variable controller 44 varies the hyperplane of theadaptive sliding mode control process in the following manner:

The hyperplane variable controller 44 determines the value of the linearfunction σ from the estimated state quantities x₁ hat, x₂ hat accordingto the equation (13) using the present value of the coefficient k, anddetermines the value of a parameter f which is defined according to theequation (21), given below, depending on the magnitude of the absolutevalue |σ| of the determined linear function σ. ##EQU16## where σ_(limit)represents a predetermined threshold for determining whether the linearfunction σ corresponding to the present estimated state quantities x₁hat, x₂ hat is substantially in agreement with the hyperplane (σ=0) ornot, i.e., whether the estimated state quantities x₁ hat, x₂ hat hassubstantially converged onto the hyperplane (σ=0) or not.

The hyperplane variable controller 44 integrates the parameter f thusdetermined in each cycle time of the adaptive sliding mode controlprocess, and determines an integrated value sum(f) as indicated by thefollowing equation (22):

    sum(f)=∫.sub.0.sup.t fdt                              (22)

and determines the present value of the coefficient k from theintegrated value sum(f) according to the following equation (23):

    k=k.sub.0 +γ·sum(f)                         (23)

where k₀ represents an initial value (>0) of the coefficient k whichdefines the hyperplane, and γ represents a predetermined gaincoefficient for adjusting the rate of change of the value of thecoefficient k. The initial value k₀ is selected such that the estimatedstate quantities x₁ hat, x₂ hat will converge onto the hyperplane (σ=0)within a shortest time.

The hyperplane variable controller 44 gives the value of the coefficientk thus determined according to the equation (23) as the value of thecoefficient k for effecting the aforesaid operations and decision to theequivalent control input calculator 39, the nonlinear input calculator40, and the stability decision unit 42.

In order to prevent the value of the coefficient k from becomingnegative and also from becoming smaller than the initial value k₀, ifthe integrated value sum(f) according to the equation (22) is smallerthan "0" (sum(f)<0), then the integrated value sum(f) is forcibly set to"0" to determine the correction coefficient k (in this case k=k₀). Ifthe value of the coefficient k is excessively large, then it cannotquickly be reduced when it is to be reduced. To avoid such a defect, ifthe integrated value sum(f) determined according to the equation (22)becomes larger than a predetermined value α, then the integrated valuesum(f) in the equation (23) is forcibly set to "α" to determine thecorrection coefficient k (in this case, k=k₀ +α=an upper limit for thecoefficient k).

Insofar as the estimated state quantities x₁ hat, x₂ hat have notconverged onto the hyperplane (σ=0), the value of the parameter f in thecoefficient k fluctuates in the vicinity of the initial value k₀. Withthe initial value k₀ established as described above, it is possible toconverge the estimated state quantities x₁ hat, x₂ hat onto thehyperplane (σ=0) substantially within a shortest time. When theestimated state quantities x₁ hat, x₂ hat have converged onto thehyperplane (σ=0), since the value of the parameter f is substantiallysteadily fixed to "1", the value of the coefficient k graduallyincreases. Consequently, when the estimated state quantities x₁ hat, x₂hat have substantially converged onto the hyperplane (σ=0), thehyperplane used in the adaptive sliding control process according to thepresent embodiment has its gradient progressively increased as shown inFIG. 10, thus increasing the stability of convergence of the estimatedstate quantities x₁ hat, x₂ hat toward the target values "q", "0" (thebalanced point on the hyperplane) and allowing the estimated statequantities x₁ hat, x₂ hat to converge toward the target values "q", "0"in a short time (i.e., with an increased response). Progressivelyincreasing the value of the coefficient k is equivalent to moving thepole--k of the control system toward a stable region in a negativedirection on the real axis Re on the complex plane shown in FIG. B.

The manner in which the value of the coefficient k is established tovary the hyperplane is not limited to the process described above, butmay be modified in various ways. For example, combinations of "2" and"-1", "1" and "-2", etc. other than "1" and "-1", may be used as thevalue of the parameter f in the equation (21). Using such combinations,it is possible to vary the rate of change of the hyperplane depending onwhether the value of the linear function σ substantially coincides withthe hyperplane (σ=0) or not. Alternatively, the value of the parameter fmay be given as a function of the value of the linear function σ to varythe hyperplane depending on the value of the linear function σ. Furtheralternatively, it is also possible to vary the value of γ in theequation (23) depending on the direction in which the value determinedby the equation (22) varies, i.e., increases or decreases, or to varythe value of γ depending on the value of the linear function σ. Anoptimum one of those processes for varying the hyperplane can beselected depending on the object to be controlled, and a specificprocess of determining the coefficient k may be determined throughexperimentation or the like in view of the stability of control or thequick response capability.

The details of the adaptive sliding mode control process have beendescribed above.

The adaptive controller 23 of the general feedback controller 20 shownin FIG. 1 will be described below.

As shown in FIG. 1, the general feedback controller 20 effects afeedback control process to converge the air-fuel ratio (the pre-CATA/F) at the LAF sensor 6 toward the target air-fuel ratio KCMD which isdetermined by the target air-fuel ratio calculator 16 as describedabove. If such a feedback control process were carried out under theknown PID control, it would be difficult keep stable controllabilityagainst dynamic behavioral changes including changes in the operatingconditions of the internal combustion engine 1, characteristic changesdue to aging of the internal combustion engine 1, etc.

According to the present embodiment, the general feedback controller 20includes, in addition to the PID controller 22 for carrying out theknown PID control process, the adaptive controller 23 for compensatingfor such dynamic behavioral changes, and switches between the feedbackcorrection coefficients KFB produced respectively by the PID controller22 and the adaptive controller 23 for performing a feedback controlprocess.

As shown in FIG. 11, the adaptive controller 23 comprises a parameteradjuster 45 for establishing a plurality of adaptive parameters usingthe parameter adjusting law proposed by I. D. Landau, et al., and acorrection coefficient calculator 46 for calculating the feedbackcorrection coefficient KSTR using the established adaptive parameters.

The parameter adjuster 45 will be described below. According to theparameter adjusting law proposed by I. D. Landau, et al., whenpolynomials of the denominator and the numerator of a transfer functionB(Z⁻¹)/A(Z⁻¹) of a discrete-system object to be controlled are generallyexpressed respectively by equations (24), (25), given below, an adaptiveparameter θ hat (j) (j indicates the number of a control cycle)established by the parameter adjuster 45 is represented by a vector(transposed vector) according to the equation (26) given below. An inputζ(j) to the parameter adjuster 45 is expressed by the equation (27)given below. In this embodiment, it is assumed that the internalcombustion engine 1, which is an object to be controlled by the feedbackcontroller 20, is considered to be a plant of a first-order systemhaving a dead time d_(P) corresponding to three control cycles (a timecorresponding to three combustion cycles of the internal combustionengine 1), and m=n=1, d_(p) =3, and five adaptive parameters s₀, r₁, r₂,r₃, b₀ are established (see FIG. 11). In the upper and middleexpressions of the equation (27), u_(s), y_(s) generally represent acontrol input (manipulating quantity) to the object to be controlled andan output (controlled quantity) from the object to be controlled. Sincethe control input is the feedback correction coefficient KSTR and theoutput from the object (the internal combustion engine 1) is the pre-CATA/F (hereinafter referred to as "KACT") actually detected by the LAFsensor 6 in the illustrated embodiment, the input ζ(j) to the parameteradjuster 45 is expressed by the lower expression of the equation (27)(see FIG. 11). ##EQU17##

The adaptive parameter θ hat expressed by the equation (26) is made upof a scalar quantity element b₀ hat ⁻¹ (j) for determining the gain ofthe adaptive controller 23, a control element B_(R) hat (Z⁻¹,j)expressed using a manipulating quantity, and a control element S hat(Z⁻¹,j) expressed using a controlled quantity, which are expressedrespectively by the following equations (28)-(30) (see the block of thecorrection coefficient calculator 46 shown in FIG. 11): ##EQU18##

The parameter adjuster 45 establishes coefficients of the scalarquantity element and the control elements, described above, and suppliesthem as the adaptive parameter θ hat expressed by the equation (26) tothe correction coefficient calculator 46. The parameter adjuster 45calculates the adaptive parameter θ hat so that the pre-CAT A/F willagree with the target air-fuel ratio, using the feedback correctioncoefficient KSTR which is a manipulating quantity from the present tothe past and the pre-CAT A/F (=KACT) which is a controlled quantity.

Specifically, the parameter adjuster 45 calculates the adaptiveparameter θ hat according to the following equation (31):

    θ(j)=θ(j-1)+Γ(j-1)·ζ(j-d.sub.p)·e*(j)                                                        (31)

where Γ(j) represents a gain matrix (m+n+d_(p)) for determining a rateof establishing the adaptive parameter θ hat, and e*(j) an estimatederror of the adaptive parameter θ hat. Γ(j) and e*(j) are expressedrespectively by the following recursive formulas (32), (33): ##EQU19##where D(Z⁻¹) represents an asymptotically stable polynomial foradjusting the convergence. In this embodiment, D(Z⁻¹)=1.

Various specific algorithms are obtained depending on how λ₁ (j), λ₂ (j)in the equation (33) are selected. For example, if λ₁ (j)=1 and λ₂ (j)=λ(0<λ<2), then a degressive gain algorithm (a method of least squares ifλ=1) is obtained. If λ₁ (j)=λ₁ (0<λ₁ <1) and λ₂ (j)=λ₂ (0<λ₂ <λ), then avariable gain algorithm (a method of weighted least squares if λ₂ =1) isobtained. When λ₁ (j)/λ₂ (j)=η and λ₃ is expressed by the equation (34),given below, if λ₁ (j)=λ₃, then a fixed trace algorithm is obtained. Inthe equation (34), "trΓ(0)" represents a trace function of a matrix Γ(0)and is the sum (scalar quantity) of diagonal elements of the matrixΓ(0). If λ₁ (j)=1 and λ₂ (j)=0, a fixed gain algorithm is obtained. Inthis case, as can be seen from the equation (32), Γ(j)=Γ(j-1), and henceΓ(j) is of a fixed value. Any one of the degressive gain algorithm, thevariable gain algorithm, the fixed gain algorithm, and the fixed tracealgorithm is suitable for a time-dependent plant such as a fuelinjection process, an air-fuel ratio, or the like of the internalcombustion engine 1. ##EQU20##

Using the adaptive parameter θ hat (s₀, r₁, r₂, r₃, b₀) established bythe parameter adjuster 45 and the target air-fuel ratio KCMD calculatedby the target air-fuel ratio calculator 16, the correction coefficientcalculator 46 determines the feedback correction coefficient KSTRaccording to the recursive formula (35) given below. ##EQU21## where"d'" represents a dead time until the pre-CAT A/F corresponding to thetarget air-fuel ratio KCMD is detected by the LAF sensor 6. In thisembodiment, the dead time d' is a time (=4·d_(p)) corresponding to 12cycles each equal to a crankshaft angle period (so-called TDC).

As is apparent from the foregoing description, the adaptive controller23 thus constructed is a recursive-type controller taking into accountdynamic behavioral changes of the internal combustion engine 1 which isan object to be controlled. Stated otherwise, the adaptive controller 23is a controller described in a recursive form to compensate for dynamicbehavioral changes of the internal combustion engine 1, and moreparticularly a controller having a recursive-type adaptive parameteradjusting mechanism.

A recursive-type controller of this type may be constructed using anoptimum regulator. In such a case, however, it generally has noparameter adjusting mechanism.

The details of the adaptive controller 23 have been described above.

The PID controller 22, which is provided together with the adaptivecontroller 23 in the general feedback controller 20, calculates aproportional term (P term), an integral term (I term), and a derivativeterm (D term) from the difference between the pre-CAT A/F detected bythe LAF sensor 6 and the target air-fuel ratio KCMD, and calculates thetotal of those terms as the feedback correction coefficient KLAF, as isthe case with the general PID control process. In this embodiment,because the fuel injection quantity is corrected by being multiplied bythe feedback correction coefficient KLAF, the feedback correctioncoefficient KLAF is "1" when the difference between the pre-CAT A/F andthe target air-fuel ratio KCMD is "0". Therefore, the integral term (Iterm) has an initial value of "1". The gains of the proportional term,the integral term, and the derivative term are determined from therotational speed and intake pressure of the internal combustion engine 1using a predetermined map.

The switcher 24 of the general feedback controller 20 outputs thefeedback correction coefficient KLAF determined by the PID controller 22as the feedback correction coefficient KFB for correcting the fuelinjection quantity if the combustion in the internal combustion engine 1tends to be unstable as when the temperature of the coolant of theinternal combustion engine 1 is low, the internal combustion engine 1rotates at high speeds, or the intake pressure is low, or if theair-fuel ratio KACT detected by the LAF sensor 6 is not reliable due toa response delay of the LAF sensor 6 as when the target air-fuel ratioKCMD changes largely or immediately after the air-fuel ratio feedbackcontrol process has started, or if the internal combustion engine 1operates highly stably as when it is idling and hence no high-gaincontrol process by the adaptive controller 23 is required. Otherwise,the switcher 24 outputs the feedback correction coefficient KSTRdetermined by the adaptive controller 23 as the feedback correctioncoefficient KFB for correcting the fuel injection quantity. This isbecause the adaptive controller 23 effects a high-gain control processand functions to converge the pre-CAT A/F detected by the LAF sensor 6quickly toward the target air-fuel ratio KCMD, and if the feedbackcorrection coefficient KSTR determined by the adaptive controller 23 isused when the combustion in the internal combustion engine 1 is unstableor the air-fuel ratio KACT detected by the LAF sensor 6 is not reliable,then the air-fuel ratio control process tends to be unstable.

The operation of the switcher 24 is disclosed in detail in Japanesepatent application No. 7-227303 (Japanese laid-open patent publicationNo. 8-105345 which corresponds to U.S. Pat. No. 5,558,075), and will notbe described in detail below.

Overall operation of the air-fuel ratio control system according to theabove embodiment will be described below.

First, a process of calculating an output fuel injection quantity #nTout(n=1, 2, 3, 4) for each of the cylinders of the internal combustionengine 1 will be described below with reference to FIGS. 1 and 13. Thecontrol unit 8 calculates an output fuel injection quantity #nTout (n=1,2, 3, 4) for each of the cylinders in synchronism with a crankshaftangle period of the internal combustion engine 1 as follows:

Outputs from various sensors including the LAF sensor 6 and the O₂sensor 7 are read in STEP13-1. The basic fuel injection quantitycalculator 12 corrects a fuel injection quantity corresponding to therotational speed and intake pressure of the internal combustion engine 1with the effective opening area of the throttle valve, therebycalculating a basic fuel injection quantity Tim in STEP13-2. The firstcorrection coefficient calculator 13 calculates a first correctioncoefficient KTOTAL depending on the coolant temperature and the amountby which the canister is purged in STEP13-3. The basic air-fuel ratiosetting unit 15 establishes a basic air-fuel ratio KBS depending on therotational speed of the internal combustion engine 1 and the intakepressure thereof indicative of a load on the internal combustion engine1 in STEP13-4.

Then, the target air-fuel ratio calculator 16 reads a basic air-fuelratio correction quantity u_(s1) which has been calculated by theadaptive sliding mode controller 19 and stored in the non-illustratedmemory in STEP13-5, and adds the basic air-fuel ratio correctionquantity u_(s1) to the basic air-fuel ratio KBS established in STEP13-4,thereby correcting the basic air-fuel ratio KBS into a target air-fuelratio KCMD in STEP13-6.

In the local feedback controller 21, the PID controllers 27 calculaterespective feedback correction coefficients #nKLAF in order to eliminatevariations between the cylinders, based on actual air-fuel ratios #nA/Fof the respective cylinders which have been estimated from the outputsignal of the LAF sensor 6 by the observer 26, in STEP13-7. Then, thegeneral feedback controller 20 calculates a feedback correctioncoefficient KFB in STEP13-8.

The general feedback controller 20 calculates a feedback correctioncoefficient KFB according to a flowchart shown in FIG. 14, using thesensor outputs read in STEP13-1 and the target air-fuel ratio KCMDdetermined in STEP13-6. Specifically, as shown in FIG. 14, the adaptivecontroller 23 and the PID controller 22 determine respective feedbackcorrection coefficients KSTR, KLAF for converging the pre-CAT A/Fdetected by the LAF sensor 6 toward the target air-fuel ratio KCMD inSTEP14-1, STEP14-2. Depending on whether the combustion in the internalcombustion engine 1 or the air-fuel ratio detected by the LAF sensor 6tends to be unstable, the switcher 24 determines whether the internalcombustion engine 1 operates in an adaptive control region which demandsan adaptive control process or not in STEP14-3. If in the adaptivecontrol region, then the switcher 24 outputs the feedback correctioncoefficient KSTR determined by the adaptive controller 23 as a feedbackcorrection coefficient KFB for correcting the fuel injection quantity ofthe internal combustion engine 1 in STEP14-4. If not in the adaptivecontrol region, i.e., if in a PID control region, then the switcher 24outputs the feedback correction coefficient KLAF determined by the PIDcontroller 22 as a feedback correction coefficient KFB for correctingthe fuel injection quantity of the internal combustion engine 1 inSTEP14-4.

When switching the feedback correction coefficient KFB from the feedbackcorrection coefficient KLAF to the feedback correction coefficient KSTR,the adaptive controller 23 determines a feedback correction coefficientKSTR in a manner to hold the correction coefficient KFB (=KSTR) to thepreceding correction coefficient KFB (=KLAF) as long as in the presentcycle time. When switching the feedback correction coefficient KFB fromthe feedback correction coefficient KSTR to the feedback correctioncoefficient KLAF, the PID controller 22 calculates a present correctioncoefficient KLAF in a manner to regard the feedback correctioncoefficient KLAF determined by itself in the preceding cycle a time asthe preceding correction coefficient KFB (=KSTR).

Referring back to FIG. 13, after the feedback correction coefficient KFBhas been calculated, the second correction coefficient calculator 14calculates in STEP13-9 a second correction coefficient KCMDM dependingon the target air-fuel ratio KCMD determined in STEP13-6.

Then, the control unit 8 multiplies the basic fuel injection quantityTim, determined as described above, by the first correction coefficientKTOTAL, the second correction coefficient KCMDM, the feedback correctioncoefficient KFB, and the feedback correction coefficients #nKLAF of therespective cylinders, determining output fuel injection quantities#nTout of the respective cylinders in STEP13-10. The output fuelinjection quantities #nTout are then corrected for accumulated fuelparticles on intake pipe walls of the internal combustion engine 1 bythe fuel accumulation corrector 28 in STEP13-11. The corrected outputfuel injection quantities #nTout a reapplied to the non-illustrated fuelinjectors of the internal combustion engine 1 in STEP13-12.

In the internal combustion engine 1, the fuel injectors inject fuel intothe respective cylinders according to the respective output fuelinjection quantities #nTout.

The above calculation of the output fuel injection quantities #nTout andthe fuel injection of the internal combustion engine 1 are carried outin successive cycle times synchronous with the crankshaft angle periodof the internal combustion engine 1 for controlling the operatingconditions thereof in order to converge the pre-CAT A/F detected by theLAF sensor 6 toward the target air-fuel ratio KCMD calculated by thetarget air-fuel ratio calculator 16. While the feedback correctioncoefficient KSTR determined by the adaptive controller 23 is being usedas the feedback correction coefficient KFB, the pre-CAT A/F is quicklyconverged toward the target air-fuel ratio KCMD with high stabilityagainst behavioral changes such as changes in the operating conditionsof the internal combustion engine 1 or characteristic changes thereof.

The basic air-fuel ratio correction quantity u_(s1) read and stored inSTEP13-5 is determined in each of cycle times of a predeterminedconstant period according to a flowchart shown in FIG. 15.

As shown in FIGS. 6, 9, and 15, after outputs from the LAF sensor 6 andthe O₂ sensor 7 are read in STEP15-1, the state predictor 18 determinesestimated state quantities x₁ hat, x₂ hat (an estimated value of thepost-CAT A/F and an estimated value of a change or rate of change of thepost-CAT A/F) of the post-CAT A/F after the dead time d in the objectexhaust system A according to the equations (2), (3) in STEP15-2.

Then, in the adaptive sliding mode controller 19, the hyperplanevariable controller 44 establishes a value of the coefficient k inSTEP15-3, and thereafter the equivalent control input calculator 39calculates an equivalent control input u_(eq) according to the equation(17) in STEP15-4. The nonlinear input calculator 40 then calculates avalue of the linear function σ according to the equation (13) inSTEP15-5, and calculates an adaptive control input u_(adp) (adaptivecontrol law term) according to the equation (19) in STEP15-6.

The nonlinear input calculator 40 compares the absolute value of thelinear function σ with a predetermined small value ε in STEP15-7. If|σ|>ε, then the nonlinear input calculator 40 calculates a reachingcontrol input u_(rch) (reaching control law term) according to theequation (18) in STEP15-8. If |σ|≦ε, i.e., if the estimated statequantities x₁ hat, x₂ hat have substantially converged onto thehyperplane, then the nonlinear input calculator 40 forcibly sets thereaching control input u_(rch) to "0" in STEP15-9.

Then, the nonlinear input calculator 40 calculates a basic air-fuelratio correction quantity u_(s1) from the equivalent control inputu_(eq), the reaching control input u_(rch), and the adaptive controlinput u_(adp) according to the equation (20) in STEP15-10.

The stability decision unit 42 determines stability of the adaptivesliding mode control process according to the flowchart shown in FIG. 16in STEP15-11. If the adaptive sliding mode control process is stable(YES in STEP15-12), then the adaptive sliding mode controller 19 outputsthe basic air-fuel ratio correction quantity u_(s1) determined inSTEP15-10 through the correction limiter 43 in STEP15-13. If theadaptive sliding mode control process is unstable (NO in STEP15-12),then the adaptive sliding mode controller 19 uses the basic air-fuelratio correction quantity u_(s1) determined in the preceding cycle timeas a present basic air-fuel ratio correction quantity u_(s1) inSTEP15-14, and outputs the basic air-fuel ratio correction quantityu_(s1) in STEP15-13. The basic air-fuel ratio correction quantity u_(s1)output in STEP15-13 is stored in the non-illustrated memory. The storedbasic air-fuel ratio correction quantity u_(s1) is read in STEP13-5shown in FIG. 13 for use in calculating the target air-fuel ratio KCMD.A process of reading the stored basic air-fuel ratio correction quantityu_(s1) will be described later on.

The basic air-fuel ratio correction quantity u_(s1) thus determined bythe adaptive sliding mode controller 19 is determined so as to convergethe post-CAT A/F detected by the O₂ sensor 7 toward the predeterminedadequate value q as described above. Therefore, the pre-CAT A/F isfeedback-controlled by the feedback controller 17 at the target air-fuelratio KCMD which has been corrected from the basic air-fuel ratio KBS bythe basic air-fuel ratio correction quantity u_(s1) for therebycontrolling the post-CAT A/F at the adequate value q under the feedbackcontrol by the feedback controller 17.

The adaptive sliding mode control process carried out by the adaptivesliding mode controller 19 has such characteristics that insofar as thestate quantities (the value of the post-CAT A/F and its change or rateof change) of the post-CAT A/F to be adjusted to the predeterminedadequate value q are converged onto the hyperplane, the state quantitiescan stably be converged toward a balanced point (a point of convergence)on the hyperplane by the equivalent control input u_(eq) without beingaffected by disturbances and a model error of the object to becontrolled. Therefore, as long as the state quantities of the post-CATA/F are converged onto the hyperplane, the post-CAT A/F can be adjustedto the adequate value q irrespective of changes in the operatingconditions of the internal combustion engine 1 and aging of thecatalytic converter 4.

In this embodiment, the adaptive sliding mode control process whichtakes disturbances, a model error, etc. into account using the adaptivecontrol law is employed for converging the state quantities of thepost-CAT A/F onto the hyperplane. Therefore, at a stage in which thestate quantities of the post-CAT A/F have not converged onto thehyperplane, the state quantities can stably be converged onto thehyperplane while assuming the effect of disturbances, a model error,etc. as being very small.

The object exhaust system A which is an object to be controlled by theadaptive sliding mode control process generally contains a relativelylong dead time d, which tends to induce control instability. Accordingto the present embodiment, however, in determining the basic air-fuelratio correction quantity u_(s1) in the adaptive sliding mode controlprocess, the state quantities of the post-CAT A/F detected on areal-time basis by the O₂ sensor 7 are not used as they are, but theestimated state quantities x₁ hat, x₂ hat produced by compensating forthe dead time d with the state predictor 18 are used. Consequently, oncethe estimated state quantities x₁ hat, x₂ hat converge onto thehyperplane, an estimation error of the estimated state quantities x₁hat, x₂ hat is absorbed due to the intrinsic characteristics of theadaptive sliding mode control process.

Therefore, the air-fuel ratio control system according to the presentembodiment can adjust the post-CAT A/F highly accurately to theappropriate value q regardless of changes in the operating conditions ofthe internal combustion engine 1, aging of the catalytic converter 4,disturbances, a model error, etc., for thereby controlling the air-fuelratio of the internal combustion engine 1 at an air-fuel ratio whichallows the catalytic converter 4 to maximize its exhaust gas purifyingcapability. As a result, an optimum emission control ability can bemaintained for the internal combustion engine 1.

Furthermore, the hyperplane variable controller 44 varies thecoefficient k which defines the hyperplane thereby to vary thehyperplane depending on the manner in which the estimated statequantities x₁ hat, x₂ hat converge onto the hyperplane. Consequently,the estimated state quantities x₁ hat, x₂ hat can converge onto thehyperplane stably within a short period of time, and even after theestimated state quantities x₁ hat, x₂ hat have converged onto thehyperplane, the estimated state quantities x₁ hat, x₂ hat can convergetoward a balanced point on the hyperplane, i.e., a point of convergencewhere x₁ hat=q and x₂ hat=0, stably within a short period of time.Therefore, the post-CAT A/F can be adjusted quickly to the appropriatevalue q within a short convergence time (with a highly quick response)and with high stability.

In this embodiment, the calculation of the output fuel injectionquantities #nTout, including the calculation of the correctioncoefficients and the calculation of the target air-fuel ratio, by thefeedback controller 17 is carried out in synchronism with the crankshaftangle period as it needs to be synchronism with the rotation of theinternal combustion engine 1. Therefore, the output fuel injectionquantities #nTout are calculated not at regular time intervals butirregular time intervals, as shown in an upper portion of FIG. 12.

The adaptive sliding mode controller 19 calculates the basic air-fuelratio correction quantity u_(s1) in successive cycle times each having agiven period CT as shown in a lower portion of FIG. 12, and stores thecalculated basic air-fuel ratio correction quantity u_(s1) in thenon-illustrated memory. The basic air-fuel ratio correction quantityu_(s1) stored in the memory is updated each time a basic air-fuel ratiocorrection quantity u_(s1) is newly determined. Thus, the basic air-fuelratio correction quantity u_(s1) is calculated and stored at times outof synchronism with the calculation of the output fuel injectionquantities #nTout. In this embodiment, the period CT at which the basicair-fuel ratio correction quantity use is calculated is longer than thecrankshaft angle period at which each of the output fuel injectionquantities #nTout is calculated.

Because the basic air-fuel ratio correction quantity u_(s1) iscalculated out of synchronism with the calculation of the output fuelinjection quantities #nTout, the target air-fuel ratio KCMD iscalculated using the basic air-fuel ratio correction quantity u_(s1) andfurthermore the output fuel injection quantities #nTout are calculatedaccording to a process described below.

As shown in FIG. 12, for calculating the target air-fuel ratio KCMD andfurthermore the output fuel injection quantities #nTout, the last basicair-fuel ratio correction quantity u_(s1) which has previously beencalculated by the adaptive sliding mode controller 19 and stored in thememory is used. If the timing of the calculation of the output fuelinjection quantities #nTout and the timing of the calculation of thebasic air-fuel ratio correction quantity u_(s1) happen to coincide witheach other, then the basic air-fuel ratio correction quantity u_(s1)which has already been stored in the memory is used to calculate theoutput fuel injection quantities #nTout, and thereafter a newlydetermined basic air-fuel ratio correction quantity u_(s1) is stored inthe memory.

Since the basic air-fuel ratio correction quantity u_(s1) and the outputfuel injection quantities #nTout are calculated in respective cycletimes independent of each other, the adaptive sliding mode controller 19and the feedback controller 17 can perform calculations in respectivecycle times which match their respective control characteristics and theobject to be controlled. In particular, the basic air-fuel ratiocorrection quantity u_(s1) is calculated by the adaptive sliding modecontroller 19 in cycle times each having a relatively long period CTcorresponding to the relatively long dead time d present in the objectexhaust system A and the response delay time thereof. Since d_(M) in theequation (3) may be constant if the cycle times are constant, any burdenon the adaptive sliding mode controller 19 for calculations can bereduced, and the adaptive sliding mode controller 19 can calculate basicair-fuel ratio correction quantity u_(s1) with high accuracy withoutcalculation errors. As a consequence, the post-CAT A/F can be adjustedto the appropriate value q highly accurately.

A simulation process effected on the air-fuel control system shown inFIG. 1 and a conventional air-fuel control system will be describedbelow.

When a disturbance L as shown in FIG. 17(a) was applied to the pre-CATA/F, the ability of the air-fuel control system shown in FIG. 1 toconverge the post-CAT A/F was simulated. The result of the simulatedpost-CAT A/F for the air-fuel control system shown in FIG. 1 is shown inFIG. 17(b). The ability of a conventional air-fuel control system, whichdetermines a basic air-fuel ratio correction quantity using theconventional PID control process, to converge the post-CAT A/F was alsosimulated. The result of the simulated post-CAT A/F for the conventionalair-fuel control system is shown in FIG. 17(c).

In the air-fuel control system shown in FIG. 1, as can be seen from FIG.17(b), the post-CAT A/F was adjusted highly accurately to theappropriate value q within a short period of time independently of thedisturbance L.

In the conventional air-fuel control system, as can be seen from FIG.17(c), the post-CAT A/F fluctuated across the appropriate value q anddid not converge toward the appropriate value q with high accuracy.

It can be seen from the results of the simulation process shown in FIGS.17(a) through 17(c) that the air-fuel control system according to thepresent embodiment is capable of adjusting the post-CAT A/F highlyaccurately to the appropriate value q within a short period of timeindependently of the disturbance because the adaptive sliding modecontrol process is used to calculate the basic air-fuel ratio correctionquantity.

An air-fuel control system according to another embodiment of thepresent invention will be described below with reference to FIG. 18. Theair-fuel control system shown in FIG. 18 is similar to the air-fuelcontrol system shown in FIG. 1 except for certain components. Thoseparts of the air-fuel control system shown in FIG. 18 which areidentical to those of the air-fuel control system shown in FIG. 1 aredenoted by identical reference numerals and representations, and willnot be described in detail below.

As shown in FIG. 18, the air-fuel control system according to the otherembodiment differs from the air-fuel control system shown in FIG. 1 withrespect to the general feedback controller 20. The general feedbackcontroller 20 has, in addition to the PID controller 22, the adaptivecontroller 23, and the switcher 24 which are identical to those shown inFIG. 1, dividers 47, 48 for dividing the pre-CAT A/F (=KACT) producedfrom the LAF sensor 6 through the filters 24, 25 by the target air-fuelratio KCMD calculated by the target air-fuel ratio calculator 16, i.e.,for determining a ratio KACT/KCMD between the pre-CAT A/F and the targetair-fuel ratio KCMD, and a target value setting unit 49 for establishinga target value (=1) for the ratio KACT/KCMD. For determining the ratioKACT/KCMD with the dividers 47, 48, since there is a dead time d'(expressed by the equation (35)) between the pre-CAT A/F (=KACT)produced from the LAF sensor 6 and the target air-fuel ratio KCMDcalculated by the target air-fuel ratio calculator 16, the dividers 47,48 are supplied with the target air-fuel ratio KCMD through a timeadjuster 50 which adjusts the dead time d'.

The ratio KACT/KCMD determined by the dividers 47, 48 is supplied to thePID controller 22 and the adaptive controller 23, and the target value(=1) for the ratio KACT/KCMD is supplied from the target value settingunit 49 to the PID controller 22 and the adaptive controller 23. The PIDcontroller 22 and the adaptive controller 23 determine the respectivefeedback correction coefficients KLAF, KSTR so as to equalize the ratioKACT/KCMD with the target value (=1). The adaptive controller 23determines the feedback correction coefficient KSTR according to arecursive formula that is similar to the equation (35) except that"KCMD(j-df)," and "KACT(j)" are replaced respectively with "1" and"KACT/KCMD".

Other details of the air-fuel control system shown in FIG. 18 areidentical to those of the air-fuel control system shown in FIG. 1.

In the air-fuel control system shown in FIG. 18, the general feedbackcontroller 20 of the above structure determines the feedback correctioncoefficient KFB (=KLAF or KSTR) such that the ratio KACT/KCMD betweenthe target air-fuel ratio KCMD corrected according to the adaptivesliding mode control process and the pre-CAT A/F detected by the LAFsensor 6 will be equalized to "1", i.e., the target air-fuel ratio KCMDwill be equalized to the pre-CAT A/F. Consequently, the air-fuel controlsystem shown in FIG. 18 offers the same advantages as those of theair-fuel control system shown in FIG. 1. Furthermore, since the targetvalue used for the general feedback controller 20 to determine thefeedback correction coefficient KFB is fixed to "1", the control processof the general feedback controller 20 is stabler than if the targetair-fuel ratio KCMD (which varies from time to time) is used as a targetvalue as is the case with the air-fuel control system shown in FIG. 1.Especially, the adaptive controller 23 of the general feedbackcontroller 20 is made much stabler because changes in the adaptiveparameter θ are reduced by the fixed target value.

In the embodiment shown in FIG. 18, the ratio KACT/KCMD between thetarget air-fuel ratio KCMD and the pre-CAT A/F detected by the LAFsensor 6 is converged toward the target value of "1". However, adifference between the target air-fuel ratio KCMD and the pre-CAT A/Fdetected by the LAF sensor 6 may be determined, and the general feedbackcontroller 20 may operate to eliminate the difference (with a targetvalue for the difference being set to "0"). Moreover, the detectedpre-CAT A/F may be corrected directly by the output u_(s1) of theadaptive sliding mode controller 19, and the general feedback controller20 may operate to equalize the corrected pre-CAT A/F to a separatelyestablished target value.

In each of the above embodiments, the wide-range air-fuel ratio sensor(LAF sensor) 6 is used as the first exhaust gas sensor. However, thefirst exhaust gas sensor may comprise an ordinary O₂ sensor or any ofvarious other sensors provided it can detect the air-fuel ratio of anexhaust gas.

Furthermore, in each of the above embodiments, the oxygen concentrationsensor (O₂ sensor) is used as the second exhaust gas sensor. However,the second exhaust gas sensor may comprise any of various other sensorsprovided it can detect the concentration of a certain component of anexhaust gas downstream of the catalytic converter. For example, ifcarbon monoxide (CO) contained in an exhaust gas downstream of thecatalytic converter is to be controlled, then a CO sensor may be used asthe second exhaust gas sensor. If nitrogen oxides (NOx) contained in anexhaust gas downstream of the catalytic converter are to be controlled,then an NOx sensor may be used as the second exhaust gas sensor. Ifhydrocarbon (HC) contained in an exhaust gas downstream of the catalyticconverter is to be controlled, then an HC sensor may be used as thesecond exhaust gas sensor. If a three-way catalytic converter is used,then the concentration of either one of the gas components describedabove may be detected to maximize the exhaust gas purifying capabilityof the catalytic converter. If a reducing or oxidizing catalyticconverter is used, then its exhaust gas purifying capability can beincreased by directly detecting a gas component to be purified.

While the air-fuel ratio of the internal combustion engine is controlledaccording to the sliding mode control process in each of the aboveembodiments, the present invention is not limited to the control of theair-fuel ratio of the internal combustion engine, but is also applicableto the control of state quantities of any arbitrary object to becontrolled.

Although certain preferred embodiments of the present invention havebeen shown and described in detail, it should be understood that variouschanges and modifications may be made therein without departing from thescope of the appended claims.

What is claimed is:
 1. A sliding mode control method comprising thesteps of:establishing a hyperplane for a sliding mode control processwith a linear function having as variables a plurality of statequantities of an object to be controlled; converging the statequantities onto said hyperplane; converging the state quantities towarda balanced point on said hyperplane while converging the statequantities onto said hyperplane, therein to control the state quantitiesat target state quantities represented by said balanced point; andvariably establishing said hyperplane depending on a value of saidlinear function.
 2. A sliding mode control method according to claim 1,further comprising the steps of:determining whether the state quantitiesare substantially converged onto said hyperplane; and gradually varyingsaid hyperplane in order to increase the stability of convergence of thestate quantities toward said balanced point or shorten the time requiredfor the state quantities to converge toward said balanced point if saidstate quantities are substantially converged onto said hyperplane.
 3. Asliding mode control method according to claim 2, further comprising thestep of:establishing the hyperplane in order to substantially minimizethe time required for the state quantities to converge onto saidhyperplane if said state quantities are not substantially converged ontosaid hyperplane.
 4. A sliding mode control method according to claim 2or 3, further comprising the step of:comparing the value of said linearfunction with a predetermined value to determine whether said statequantities are substantially converged onto said hyperplane or not.
 5. Asliding mode control method according to claim 2, wherein said statequantities comprise two state quantities, further comprising the stepof:gradually increasing the gradient of said hyperplane if said statequantities are substantially converged onto said hyperplane.
 6. Asliding mode control method according to claim 5, further comprising thesteps of:establishing a predetermined positive numerical value and apredetermined negative numerical value in each predetermined cycle timedepending on whether said state quantities are substantially convergedonto said hyperplane or not; integrating the established numericalvalues; and establishing said hyperplane by varying the gradient of thehyperplane from a predetermined initial value of the gradient by anamount corresponding to the integrated value.
 7. A sliding mode controlmethod according to claim 6, wherein said initial value of the gradientis determined to substantially minimize the time required for the statequantities to converge onto said hyperplane if said state quantities arenot substantially converged onto said hyperplane.
 8. A sliding modecontrol method according to claim 6 or 7, further comprising the stepof:varying the gradient of said hyperplane within a predetermined range.9. A sliding mode control method according to claim 1 or 2, furthercomprising the step of:converging said state quantities onto saidhyperplane according to an adaptive sliding control mode includingreaching and adaptive control laws.
 10. A sliding mode control methodcomprising the steps of:establishing a hyperplane for a sliding modecontrol process with a linear function having as variables a pluralityof state quantities of an object to be controlled, wherein said statequantities comprise two state quantities; converging the statequantities onto said hyperplane; converging the state quantities towarda balanced point on said hyperplane while converging the statequantities onto said hyperplane, therein to control the state quantitiesat target state quantities represented by said balanced point; andvariably establishing said hyperplane depending on the manner in whichthe state quantities are conveyed onto said hyperplane; determiningwhether the state quantities are substantially converged onto saidhyperplane; and gradually varying said hyperplane in order to increasethe stability of convergence of the state quantities toward saidbalanced point or shorten the time required for the state quantities toconverge toward said balanced point if said state quantities aresubstantially converged onto said hyperplane; gradually increasing thegradient of said hyperplane if said state quantities are substantiallyconverged onto said hyperplane; establishing a predetermined positivenumerical value and a predetermined negative numerical value in eachpredetermined cycle time depending on whether said state quantities aresubstantially converged onto said hyperplane or not; integrating theestablished numerical values; and establishing said hyperplane byvarying the gradient of the hyperplane from a predetermined initialvalue of the gradient by an amount corresponding to the integratedvalue.
 11. A sliding mode control method according to claim 10, whereinsaid initial value of the gradient is determined to substantiallyminimize the time required for the state quantities to converge ontosaid hyperplane if said state quantities are not substantially convergedonto said hyperplane.
 12. A sliding mode control method according toclaims 10 or 11, further comprising the step of:varying the gradient ofsaid hyperplane within a predetermined range.